Question: Given $ m \angle QPR = 7x - 57$, $ m \angle RPS = 6x + 34$, and $ m \angle QPS = 159$, find $m\angle RPS$. $P$ $Q$ $S$ $R$
Solution: From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Substitute in the expressions that were given for each measure: $ {7x - 57} + {6x + 34} = {159}$ Combine like terms: $ 13x - 23 = 159$ Add $23$ to both sides: $ 13x = 182$ Divide both sides by $13$ to find $x$ $ x = 14$ Substitute $14$ for $x$ in the expression that was given for $m\angle RPS$ $ m\angle RPS = 6({14}) + 34$ Simplify: $ {m\angle RPS = 84 + 34}$ So ${m\angle RPS = 118}$.